Field of the Invention
This invention relates to multimode optical fibers (MMFs) and, more particularly, to the design and manufacture of such fibers optimized for broadband applications, including coarse wavelength division multiplexing (CWDM).
Discussion of the Related Art
A typical MMF includes a relatively high-index core region surrounded by a lower index cladding region, with the two regions configured to support the simultaneous propagation of optical radiation in the core region in a plurality of transverse modes. The base material of MMFs is typically silica glass, with the core region being up-doped with one or more dopants (e.g., Ge, Al, P) that increase its refractive index and the cladding region being either undoped or down-doped with one or more dopants (e.g., F, B) that reduce its refractive index. In some designs, dopants such as F or B may also be added to the core region as long as the net refractive index of the core region is still greater than that of the cladding region.
The choice of a specific dopant (and its concentration profile) in the core and cladding regions may be dictated by design characteristics (e.g., index grading, NA, MFD) or performance issues (e.g., bandwidth), or may dictated by manufacturing/process problems associated with the use of a particular dopant (e.g., P).
More specifically, Ge-dopant is commonly used to form a near-parabolic index profile in the core region of a MMF. While the Ge-doped index profile in a MMF can be optimized to achieve a high bandwidth, the high material dispersion of Ge-doped silica limits the spectral width of the high bandwidth region. It is known that both P and F doped silica have much smaller material dispersion relative to Ge-doped silica, and fibers made with P- and/or F-dopants have much wider spectral width than conventional Ge-doped fiber [1]. However, it is difficult to introduce a high P-dopant concentration during preform processing because P-doped silica has a high vapor pressure, and a significant fraction of P-dopant is burned off during preform collapse. It is also difficult to maintain a circular preform core containing a high P-concentration because of its much lower viscosity than the surrounding silica substrate tube.
Furthermore, upon exposure to either hydrogen or radiation, fibers containing a high P-concentration have a significantly higher added attenuation, and the added attenuation increases monotonically with the P-dopant concentration. Therefore, it is desirable to limit the P-concentration in the fiber core region.
Thus, a combination of dopants including but not limited to Ge, P, Al, B, and F is required to satisfy both the material dispersion properties imposed by the required broadband (e.g., CWDM) operation as well as to resolve the above manufacturing issues. In the prior art, MMFs have been analyzed and designed using the so-called “α-profile” to reduce modal dispersion, where the refractive index profile shape of the core region is substantially parabolic. Such a procedure may be too restrictive to achieve effective CWDM-optimized MMFs while at the same time addressing process/manufacturing issues.
Modal dispersion is a significant impairment that limits the bit rates and/or the reach of MMF links. Grading the refractive index profile by slowing down the modes propagating along the fiber axis compared to the skew modes mitigates modal dispersion. Such fibers can be mathematically described by their refractive index n(r):
                              n          ⁡                      (            r            )                          =                  {                                                                                                                                                                                              n                            1                                                    ⁡                                                      [                                                          1                              -                                                              2                                ⁢                                                                                                      Δ                                    ⁡                                                                          (                                                                              r                                        a                                                                            )                                                                                                        α                                                                                                                      ]                                                                                                    1                          /                          2                                                                    ,                                                                                                  r                      ≤                      a                                                                                                                                                          n                        0                                            ,                                                                                                  r                      ≥                      a                                                                                  ⁢                                                          ⁢              Δ                        =                                                            n                  1                  2                                -                                  n                  0                  2                                                            2                ⁢                                  n                  1                  2                                                                                        (        1        )            where n1 is the refractive index at the center of the core region, and no is the refractive index at the core region radius r=a. The parameter α describes the index grading with α=2 corresponding to a parabolic profile and α→∞ resulting in a step-index profile. Analysis of the propagation characteristics of such graded index fibers by Gloge and Maractili [1] suggested that the optimum α is given by α=(2−2Δ). However, subsequent analysis by Keck and Olshansky [2] showed that the optimum α is given by:
                    α        =                  2          +          y          -                                                    Δ                ⁡                                  (                                      4                    +                    y                                    )                                            ⁢                              (                                  3                  +                  y                                )                                                    (                              5                +                                  2                  ⁢                  y                                            )                                                          (        2        )            where y, the profile dispersion parameter, characterizes the dispersion difference between the core and the cladding and is defined by:
                    y        =                              -                                          2                ⁢                                  n                  1                                                            N                1                                              ·                      λ            Δ                    ·                      dΔ            dλ                                              (        3        )            where λ is the wavelength of signal light propagating in the core region and N1=n1−λdn1/dλ is the group index at the center of the core region. We emphasize that the optimization metric in the Keck and Olshansky approach was the RMS pulse-width at the given wavelength assuming that all mode groups are equally excited. The pure α-profile multimode fiber, equation (1), is typically achieved by doping silica with an appropriate dopant such as GeO2 or P2O5 or F or B2O3 etc. The dopant mix employed can result in variation of the profile dispersion parameter with wavelength. As a consequence, graded index fibers optimized at one wavelength as per Keck and Olshansky's approach can have considerable performance degradation at other wavelengths.
Before we discuss prior art related to the design of wideband (i.e., broadband) MMFs, we note that another consequence of the dopant mix used in α-profile fibers is that the profile dispersion parameter y can vary radially. However, Keck and Olshansky had assumed that the y-parameter is constant across the core region. This discrepancy was noted by Marcatili and a more general class of fibers with arbitrary profile dispersion, that depends on the radius, was analyzed [3],[4]:n2(r,λ)=n12[1−F(r,λ)],F(r=0,λ)=0,F(r=a,λ)=2Δ  (4)where F(r,λ) is the arbitrary profile function. Marcatili showed that if the derivatives of F(r,λ) with respect to radius and wavelength satisfy:
                              1          +                                    r                              2                ⁢                F                                      ·                                          ∂                F                                            ∂                r                                                    =                              D            ⁡                          (              λ              )                                ⁡                      [                          1              -                              ρ                2                                      ]                                              (        5        )            with the arbitrary profile dispersion parameter, ρ, defined as:
                              ρ          ⁡                      (                          r              ,              λ                        )                          =                                            n              1                                      N              1                                ·                      λ            F                    ·                                    ∂              F                                      ∂              λ                                                          (        6        )            then the group delays τ depend only on D and the propagation constant β as follows:
                              τ          =                                                    N                1                            c                        ⁡                          [                                                1                  -                                      B                    /                    D                                                                                        1                    -                    B                                                              ]                                      ,                  B          =                      1            -                                          β                2                                                              n                  1                  2                                ⁢                                  k                  0                  2                                                                    ,                              k            0                    =                                    2              ⁢              π                        λ                                              (        7        )            Marcatili chose the parameter D to minimize the worst-case modal dispersion at a particular wavelength resulting in D≈(2−Δ) [4].
Designing a MMF for operation over a wide range of wavelengths using a single α-profile requires multiple dopants. Furthermore, the required dopant concentrations can be sufficiently high to cause manufacturability problems related to viscosity, thermal expansion among others, [6]. Sometimes the feasible NA was lower than required from a light transmission perspective. In order to circumvent these problems, Olshansky proposed multiple-α refractive index profiles [5],[6]:
                              F          ⁡                      (                          r              ,              λ                        )                          =                              ∑            i            N                    ⁢                      2            ⁢                                                  ⁢                                          Δ                i                            ⁡                              (                λ                )                                      ⁢                                          (                                  r                  a                                )                                            α                i                                                                        (        8        )            
Equation (5) when applied to this profile function yields the optimum αi at any wavelength,
                                          α            i                    =                      2            -                                                            2                  ⁢                                                                          ⁢                                      n                    1                                                                    N                  1                                            ·                              λ                                  Δ                  i                                            ·                                                dΔ                  i                                dλ                                      -                                          12                ⁢                                                                  ⁢                Δ                            5                                      ,                                            ∑                              i                =                1                            N                        ⁢                          Δ              i                                =          Δ                                    (        9        )            where Olshansky chose D≈(2−6Δ/5) to minimize the RMS pulse-width. The parameters Δi are chosen to achieve wideband performance which is enforced by requiring:
                                                        dα              i                        dλ                    =          0                ,                              or            ⁢                                                  ⁢                                          α                i                            ⁡                              (                                  λ                  1                                )                                              =                                                    α                i                            ⁡                              (                                  λ                  2                                )                                      =                          …              =                                                α                  i                                ⁡                                  (                                      λ                    q                                    )                                                                                        (        10        )            Olshansky further assumes employing m dopants whose concentration Cj is given by:
                                                        C              j                        ⁡                          (              r              )                                =                                    C                              j                ⁢                                                                  ⁢                0                                      +                                          ∑                                  i                  =                  1                                N                            ⁢                                                                    C                    ji                                    ⁡                                      (                                          r                      a                                        )                                                                    α                  i                                                                    ,                  j          =          1                ,        …        ⁢                                  ,        m        ,                  0          ≤          r          ≤          a                                    (        11        )            where Cj0 is the dopant concentration at the center of the core region (r=0) and Cji is the additional concentration at the radius (r=a) of the core region. The design process involves determining these dopant concentrations such that both the wideband requirements imposed by equation (10) as well as the NA specification are satisfied. Olshansky [5],[6] and later M-J. Li et al. [7] and D. C. Bookbinder et al. [8] discuss the specific case of two dopants and two α's. In this case, the two dopant concentrations can be re-cast in the form (for 0≦r≦α):
                                          C            1                    ⁡                      (            r            )                          =                              C            10                    -                                    (                                                C                  10                                -                                  C                                      1                    ⁢                                                                                  ⁢                    a                                                              )                        ⁡                          [                                                                    (                                          1                      -                                              x                        1                                                              )                                    ⁢                                                            (                                              r                        a                                            )                                                              α                      1                                                                      +                                                                            x                      1                                        ⁡                                          (                                              r                        a                                            )                                                                            α                    2                                                              ]                                                          (        12        )                                                      C            2                    ⁡                      (            r            )                          =                              C            20                    -                                    (                                                C                  20                                -                                  C                                      2                    ⁢                                                                                  ⁢                    a                                                              )                        ⁡                          [                                                                                          x                      2                                        ⁡                                          (                                              r                        a                                            )                                                                            α                    1                                                  +                                                      (                                          1                      -                                              x                        2                                                              )                                    ⁢                                                            (                                              r                        a                                            )                                                              α                      2                                                                                  ]                                                          (        13        )            where Cja is the dopant concentration at the core region radius r=a. It should be noted that all these fiber designs rely on the original multiple-α profile theory formulated by Olshansky [5], [6] and differ only in the particular dopants considered. Specifically, Olshanky considered the case of dopant 1 being GeO2 and dopant 2 being B2O3 with CGea=0 and CBG=0 and derived an approximate set of equations to solve for x1, x2 [5],[6]. Li et al. supra considered the case of GeO2 and F without assuming C1a=0 and C20=0 [7]. Bookbinder et al. supra further considered the use of P2O5 as a co-dopant in addition to GeO2, while including trench-based bend-insensitive MMF designs. While later work by Matthijse et al. [9] refer to MMF designs with GeO2 and F, characteristics of the dopant profiles mentioned therein were already observed in profiles governed by equations (12)-(13) in the prior work by Li [7].
In contrast to the above dopant profiles being governed by single/multiple-α's, the work by Fleming and Oulundsen [10] indicates that MMF designs with non-parabolic dopant concentration profiles can result in wideband operation over the 780-1550 nm wavelength window.
The dopant concentrations in all the prior art described thus far stem from either an assumed dopant profile function consisting of single/multiple-α's or from ad hoc procedures. As a consequence, it is not clear whether the resulting dopant concentrations are indeed optimal from a wideband perspective. In fact, an adequate definition of optimality of wideband MMF designs with arbitrary profile shapes has not been properly established in the literature. While Olshansky imposes the constraint defined by equation (10), it already assumes the presence of a multiple-α dopant profile [5],[6]. On the other hand, Marcatili supra specifies the following criterion for wideband optimality for the more general class of fibers defined by equation (5):
                              dD          dλ                =        0                            (        14        )            Since the parameter D has previously been chosen to be D(λ)≈1+√{square root over (1−2Δ(λ))}≈[2−Δ(λ)] as per Marcatili [3],[4] or D(λ)≈[2−6Δ(λ)/5] as per Olshansky [5],[6], its wavelength dependence is automatically defined. Therefore, ensuring
      dD    dλ    =  0requires the fiber Δ to be wavelength independent to the first order. Once the dopant concentrations at the core region center (r=0) and at the interface between the core and cladding regions (r=a) have been chosen to satisfy the specified NA at the design wavelength, the behavior of Δ(λ) is purely a function of the material properties. Therefore, satisfying equation (14) is not feasible and hence this optimality criterion is itself suspect. Consistent with this observation, it can be shown that the optimality criterion proposed by Olshansky for multiple-α profiles, equation (10), is not in agreement with that proposed by Marcatili, equation (14).
Thus, the prior art lacks an adequate optimality criterion for wideband operation with arbitrary profile shapes and also a systematic procedure for designing MMFs that satisfy the optimality criterion. Yet another limitation of the prior art [5]-[8] related to the multiple a profile designs [Olshansky, Li, supra] is the case where the dopant concentration at the interface between the core/cladding regions (r=a) is the same as that at the core region center (r=0); i.e., C2a=C20. From equation (13), it follows that the dopant concentration C2(r)=C20; i.e., C2(r) is constant across the core region. As described below, the invention includes MMF designs where even in this case, the dopant profile does not need to be constant, yet will still satisfy the wideband optimality criteria proposed herein.